I use slightly different notation here than Ref. [1]}
to keep the correspondence with prior work. Opposed to pure states,
for mixed \(\mathcal {S}\mathcal {E}\)  states within the pure decohering scenario, Eq. (REF )
and Eq. (REF ), the error probability (bound) is \(\sqrt{p_{1}p_{2}}\prod _{k\in \mathcal {F}}\mathrm {tr}\left[\rho _{\left.k\right|1}^{1/2}\rho _{\left.k\right|2}^{1/2}\right]\) 
for both spin and photon models [2]}, [3]}
(i.e., \(c=1/2\)  is optimal). Either prefactor, \(\min \left[p_{1},p_{2}\right]\) 
or \(\sqrt{p_{1}p_{2}}\) , will give a bound for the pure state case.
Letting the prefactor to be just some \(C\) , the QCB result for pure,
homogeneous \(\mathcal {S}\mathcal {E}\)  is
\(I_{\mathrm {acc}}\left(\hat{\Pi }_{\mathcal {S}}\right)\ge H_{\mathcal {S}}-h\left(C\left|\gamma \right|^{2{^{\sharp }}\hspace{-1.99168pt}\mathcal {F}}\right)\equiv \mathcal {X}_{QCB},\) 
